Estudio técnico

Chaos Theory is a mathematical theory that can be used to explain com- plex systems such as weather. Although many complex systems appear to behave in a random manner, chaos theory shows that, in reality, there is an underlying order that is difficult to see. Many complex systems can be better understood through the lens of Chaos Theory. Henri Poincar´e laid the groundwork for Chaos Theory. He was the first to point out that many deterministic systems display a sensitive dependence on initial conditions. Later, in the 1900s, Edward Lorenz (1963, 1965) studied Chaos Theory in the context of weather systems. When making weather predictions, he no- ticed that his calculations were significantly impacted by the extent to which he rounded his numbers. The end result of the calculation was significantly different when he used a number rounded to three digits as compared to a number rounded to six digits. His observations on Chaos Theory in weather systems led to his famous talk, which he entitled, ”Predictability: does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”. Lorenz chaos model equations are:

     ˙ X =_{−σX + σY} ˙ Y =_{−XY + rX − Y} ˙ Z = XY _{− bZ}

where σ is called the Prandtl number and r is called the Rayleigh number. All σ, r, b > 0, but usually σ = 10, b=8/3 and r is varied. The parameters σ, r, b are kept constant within an integration, but they can be changed to create a family of solutions of dynamical system defined by the differential equations. The particular parameter values chosen by Lorenz (1963) were: σ = 10, b=8/3 and r=28 -which result in chaotic solutions (sensitivity dependence on the initial conditions). Results from 3-dimensional Lorenz system illustrate the dispersion of finite time integrations from an ensemble of initial conditions (Fig. 2.8). The different initial points can be considered as estimates of the ”true” state of the system (which can be thought of as any point inside the ellipsoid) and the time evolution of each of them as possible forecasts. Subject to the initial ”true” state of the system, points close together at

initial time diverge in time at different rates. Thus, depending on the point chosen to describe the system time evolution, different forecasts are obtained.

Figure 2.8: Lorenz attractor with superimposed finite-time ensemble integra- tion (source: ECMWF).

The two wings of the Lorenz attractor can be considered as identifying two different weather regimes, for example one warm and wet and the other cold and sunny. Suppose that the main purpose of the forecast is to predict whether the system is going through a regime transition. When the system is in a predictable initial state (Fig. 2.8(a)), the rate of forecast divergence is small and all the points stay close together untill the final time. Whatever

the point chosen to represent the initial state of the system, the forecast is characterised by a small error and a correct indication of a regime transition is given. The ensemble of points can be used to generate probabilistic forecast of regime transitions. In this case, since all points end in the other wing of the attractor, there is a 100% probability of regime transition. By contrast, when the system is in a less predictable state (Fig. 2.8(b)), the points stay close together only for a short time period and then start diverging. While it is still possible to predict with a good degree of accuracy the future forecast state of the system for a short time period, it is difficult to predict whether the system will go through a regime transition in the long forecast range. Fig. 2.8(c) shows an even worse scenario, with points diverging even after a short time period and ending in very distant part of the system attractor. In probabilistic terms, one could have only predicted that there is 50% chance of the system undergoing a regime transition. Morover, the ensemble of points indicates that there is a greater uncertainty in predicting the region of the system attractor where the system will be at final time in the third case (Fig. 2.8(c)).

The comparison of the points’ divergence during the three cases indicates how ensemble prediction systems can be used to ”forecast the forecast skill” (Tennekes et al., 1986). In the case of the Lorenz system, a small divergence is associated to a predictable case and confidence can be attrached to any of the single deterministic forecasts given by the single points. By contrast, a large diverge indicates low predictability.

Similar sensivity to the initial state is shown in weather prediction. Fig. 2.9 shows the forecasts for air temperature in London given by 33 different forecasts started from very similar initial conditions for two different dates, the 26th _{of June of 1995 and the 26}th_{of June of 1994; in practice the image is}

about forecasts in the same place, one year apart. There is a clear difference degree of divergence during the two cases. All forecasts stay close together up to forecast day 10 for the first case (Fig. 2.9(a)), while they all diverge already at forecast day 3 in the second case (Fig. 2.9(b)). The level of spread among the different forecast can be used as a measure of the predictability of the two atmospheric states.

Figure 2.9: ECMWF forecasts for air temperature in London started from (a) 26 June 1995 and (b) 26 June 1994 (source: ECMWF).